General Relativity/Metric tensor

Recall that a tensor is a linear function which can convert vectors into scalars. Recall also that a distance can be stated as a formula that converts vectors to a scalar. So can we express distance with tensors formulas? Yes, we can.

The first problem comes in, in that tensors are linear functions, but we have some squares in our distance formula. We can deal with this with a mathematical trick. Consider the formula for distance in normal three dimensional Euclidean space using cartesian coordinates:

Now δij{\displaystyle \delta _{ij}} is obviously a tensor. What type of tensor is it? Well it takes two contravariant vectors and turns them into a scalar ds2{\displaystyle ds^{2}}. So it must be a covariant tensor of rank 2. δij{\displaystyle \delta _{ij}} is called the Kronecker delta tensor, which is 1 whenever i=j{\displaystyle i=j} and 0 otherwise. In general, instead of components δij{\displaystyle \delta _{ij}}, we have gij{\displaystyle g_{ij}} :

where the Greek letters just remind us that we are summing over four dimensional space time. Now in the case of special relativity gμν{\displaystyle g_{\mu \nu }} is zero for where μ{\displaystyle \mu } and ν{\displaystyle \nu } are different, +1 for the space indices 1,2,3 and −c2{\displaystyle -c^{2}} for the time index. We can call this special matrix η{\displaystyle \eta }, giving us the formulas: